466- ON METHODS FOE DETECTING SMALL OPTICAL RETARDATIONS so that the brightness of the edge's is. now about 80 times that at th of the field. The remaining values of / in Table III have been calcu before with omission of the terms representing minor periodic fluctua Hitherto we have treated various kinds of screening, but without ad retardation at the plane of the first aperture. The introduction retardation is, of course, a complication, but in principle it gives ris difficulty, provided the retardation be linear in 6 over the various par aperture. The final illumination as a function of <£ can always be e? by means of the Si- and Ci-functions. As the simplest case which presents something essentially novel, suppose that an otherwise constant retardation (R) changes sign whe is equal (say) to + p when 6 is positive and to — p when 6 is negative (3) becomes _ . m i sin 6£ . 1 — cos = 2 sin T \ cos p —z— + sm p------— reducing to (5) when p = Q. This gives the vibration at the point , second aperture. If £ = 0, (30) becomes 20 cos p sin T, and vanish* cos p = 0 ; for instance, when the whole difference of retardation 2/j (reckoned in wave-lengths) %\. The vibration in direction (p behind the second aperture is to be c by writing T+ <££ for T in (30) and integrating with respect to £. Th n • rn^ j.f , ' 2 sin T I dt; cos <££ cos p — + sm p • •*- f sin (9 m <f>l--<coBp —j- ~ ( 3 l-cos#£ and the. illumination (/) is independent of the sign of </>, which we ma; forward suppose to be positive. If the second aperture be symmetrically placed, we may take the 1 be expressed as + f, and (31) becomes . . „, 2 sm T cos p /•* sin (0 .+ <6) f + sin (0 - - - -z—r. - - 6 If we apply this to £ =90 to find what occurs when there is no sc we fall upon ambiguities, for (32) becomes '7r} -f'2 cos Tsinp [2 Si (<£(•) -$TT ± ITT},s the values of 7 for various values of 0/0. Those in the 2nd and 8th columns are smoothed values as explained, and they would remain undisturbed if tfee value of 0£ were increased. It will be seen that the maximum illumination near the edges is some 6 times that at the centre.n unity. Under the same conditions the Ci's in (21) may be omitted, so that